Can I have different rewards for a single action based on which state it transitions to?
The reward function can be a function of the current state, current action, and next state: $R(s_t, a_t, s_{t+1})$. It's valid to use the Bellman operator in this setting because it's still a contraction and will yield the optimal value function.

NOTE: I'm assuming that you will be solving the MDP with the Bellman equation.